Preheat the oven to 350°F (175°C). Grease an 8-inch square baking pan or line it with parchment paper.
In a large mixing bowl, combine the mashed bananas, unsweetened applesauce, whole milk, egg, and vanilla extract. Mix well until smooth.
In a separate bowl, whisk together the all-purpose flour, whole wheat flour, baking powder, baking soda, ground cinnamon, and salt.
Gradually add the dry ingredients to the wet ingredients, stirring gently until just combined. Do not overmix.
Fold in the melted unsalted butter until evenly incorporated.
Pour the batter into the prepared baking pan and spread it evenly.
Bake for 30 to 35 minutes, or until a toothpick inserted into the center comes out clean.
Remove the cake from the oven and let it cool in the pan for 10 minutes.
Transfer the cake to a wire rack and let it cool completely before serving to toddlers.